bifurcation

Preposition: of

• equilibria: The paper provides a very readable introduction into the basic mathematical tools for studying existence and local bifurcation of relative equilibria.

Converse of object

• fold: This is shown by the fold bifurcation in the second order model.
• undergo: The sequence of diagrams above shows the stable set, denoted by the blue curve, undergoing a type-2 contact bifurcation.
• slide: The project has observed some interesting phenomena, including grazing bifurcations and possibly grazing sliding bifurcations.

Adjective modifier

• homoclinic: An analysis of this homoclinic bifurcation is performed in a general frame.
• subcritical: The first bifurcation of the vertically upward position is a " reverse " subcritical pitchfork bifurcation denoted by PF2.
• supercritical: In both cases, the stable solutions emanating from supercritical pitchfork bifurcations underwent period-doubling cascades leading to chaos.
• carotid: METHODS: Two unusual cases of primary HPT caused by an ectopic PA located at the carotid bifurcation are reported.
• global: There one finds a similar behavior in the local and the global bifurcation.

Modifies a noun

• diagram: Our results are extended by numerical methods giving a more detailed bifurcation diagram.
• theory: The onset of chaos is often studied by bifurcation theory.
• analysis: In the second half a large number of examples are provided, often with some bifurcation analysis involved.
• phenomenon: As one result we detect bifurcation phenomena which do not exist in this form for smooth systems.
• parameter: The output time constant is used as bifurcation parameter, yielding to chaos and unstable periodic orbits.

Noun used with modifier

• codimension-two: We have also shown that the eigenvalue classification scheme of [ 1 ] cannot be used to directly classify the codimension-two bifurcations.
• period-doubling: We denote these bifurcations as " reverse " period-doubling bifurcations.
• pitchfork: The equilibrium is supposed to undergo a reversible pitchfork bifurcation, controlled by the system's parameter.
• saddle-node: It also marks the change in stability along the saddle-node bifurcation curve S.
• two-parameter: Their use in a continuation setting is shown with one- and two-parameter bifurcation studies of a transmission line model.
• grazing: Stephen Foale and Steve Bishop presented their elegant study of grazing bifurcations in impacting systems, relevant to the rattling of engineering structures.